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Families of convex tilings
Volume 64, no. 1
(2022),
pp. 87–101
https://doi.org/10.33044/revuma.3127
Abstract
We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such
tilings come in continuous families obtained by moving tile edges parallel
to themselves (keeping edge directions fixed). We study how the tile
shapes and areas change in these families. In particular we show that if
$R$ is convex, the tile shapes can be arbitrarily prescribed (up to
homothety). We also show that the tile areas and tile “orientations”
determine the tiling. We associate to a tiling an underlying bipartite
planar graph $\mathcal{G}$ and its corresponding Kasteleyn
matrix $K$. If $\mathcal{G}$ has
quadrilateral faces, we show that $K$ is the differential of the map from
edge intercepts to tile areas, and extract some geometric and probabilistic
consequences.
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Published by the Unión Matemática Argentina |